"Representing ideas and connecting the representations to mathematics lies at the heart of understanding mathematics. Teachers should analyze student representations and carefully listen to their discussions to gain insights into the development of mathematical thinking and to enable them to provide support as students connect their languages to the conventional language of mathematics."

(NCTM, 2000, p. 136)

Representation is not about manipulating symbols; it is about showing the ideas in a mathematical situation in a variety of meaningful ways. When we solve mathematical problems, a core part of the solution process is how we represent the ideas in the problem. The form of representation we select allows us to manipulate the information (as opposed to manipulating symbols) to reach a sensible solution. With the Representation Standard, our goal is for young students to show their mathematical ideas and procedures in multiple ways. Through the use of representations, children develop their own mental images of mathematical ideas.

Representations used in the early grades may not be those that are traditionally used by adults. Students' representations provide a record of their efforts to understand mathematics and to make that understanding accessible to others. It is through representations that we can get into a child's thinking, assess that child's understanding, and make instructional decisions accordingly.

It should be an expectation, even in the early grades, that students use multiple representations. For example, children might use manipulative materials to model their thinking about a problem, then translate that model to a drawing on paper, and eventually move toward the use of more conventional symbols in expressing their thinking. Multiple representations support students' thinking by allowing them to see the same idea expressed in different ways. Let's take a look at some of ways that young children can represent their thinking.